Here are a number of misleading sequences. I have given the first few numbers of each sequence, and your mission, if you choose to accept it, is to guess the next number.

**1.**** A Doubling Dilemma**

1, 2, 4, 8, 16, ___?

Did you get 32 from doubling?

The answer is actually **31**. Yes, you read that correctly. 31. The sequence is the number of pieces a circle is divided into by lines connecting a number of points.

One point does not cut a circle, two points cut a circle into 2 pieces, three points into 4 pieces, four points into 8 pieces, and five points into 16 pieces. But when you get to six points, there are only 31 pieces. If you don’t believe me, count it for yourself!

The continuation: 1, 2, 4, 8, 16, 31, 57, 99, 163, ….

**2. The Broken Computer**

**1, 1, 1, 1, ___?**

Did you get 1?

The answer is *obviously* **2**. I am reading off numbers from Pascal’s Triangle, each row from left to right.

The continuation: 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, ….

**3. A Prime Sequence**

**2, 3, 5, 7, 11, 13, ___?**

Did you get 17, the next prime number?

Nope, the answer, of course, is actually **19**, the next number of fractions in a Farey sequence.

The continuation: 2, 3, 5, 7, 11, 13, 19, 23, 29, 33, ….

**4. Another Prime Sequence**

**2, 3, 5, 7, 11, 13, ___, ___?**

Wait a minute, you know better than than to guess 17 again!

Except this time, the next number actually is **17**. But the next number after that is **107**. This sequence is the minimum sum or difference of two divisors of the product of all of the previous numbers, and which is also greater than 1.

The continuation: 2, 3, 5, 7, 11, 13, 17, 107, 197, 3293, 74057, ….

**5. Elementary**

**1, 2, 3, 4, ___?**

The answer is **8**. The sequence is the number of digits of perfect numbers. A perfect number is a number whose proper divisors add up to itself. For example, 6 is perfect because 1 + 2 + 3 = 6. Also, 28 is perfect because 28 = 1 + 2 + 4 + 7 + 14.

The continuation: 1, 2, 3, 4, 8, 10, 12, 19, 37, ….

**6. Kindergarten**

**1, 2, 3, 4, 5, ___?**

The answer, of course, is **7**. This is the sequence of numbers whose divisors form an increasing geometric sequence. 6 does not work because 6 = 2×3 but 3 is not a multiple of 2. It is also the sequence of prime powers (including a prime to the 0th power).

The continuation: 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, ….

**7. Piece of Pie**

**π/2, π/2, π/2, π/2, π/2, π/2, π/2, ___?**

It is immediately clear that the answer is

**467807924713440738696537864469π/935615849440640907310521750000.**

I was informed of this sequence by a good friend of mine. It has to do with integrals of products of trigonometric functions—I am not quite sure what the continuation is, but you can read more about it here.

**Grading**

**0** – Good, you still have sanity in you.

**1 or above** – You need to see a psychiatrist.

lol

Good article

fascinating….

So good; I shall show this in my g and T lesson!